Homework 9. Due Tue. Apr.18

(1) Let U^- be the set of all (u_{ij}) in GL_n(k) such that u_{ij}=0 for i<j, u_{ii}=1 for all i.
Let B be the set of all (b_{ij}) in GL_n(k) such that b_{ij}=0 for i>j.
For any g=(g_{ij}) in GL_n(k) and any h in 1,2,...,n we define D_h(g)=det((g_{ab})_{a,b in 1,2,...,h})
Show that the following two properties are equivalent

(i) g=(g_{ij}) in GL_n(k) is of the form ub for some u in U^-,b in B.

(ii) D_h(g)=nonzero for h=1,2,...,n.

If these conditions are satisfied show that u,b in (i) are unique and that b=(b_{ij}) satisfies
b_{11}=D_1, b_{22}=D_2/D_1, b_{33}=D_3/D_2, ...

(2) Let R be the vector space of regular functions from GL_n(k) to k.

(i) Show that there is a unique linear map D:R\to R\otimes R such that
for f\in R we have D(f)= sum_{m in 1,...,s}f_m\otimes f'_m
where f_m\in R,f'_m\in R, f(xy)=sum_{m in 1,...,s}f_m(x)f'_m(y)
for any x,y in GL_n(k).

(ii) Show that the composition of a regular
function GL_n(k)\to k with x\to x^{-1} is a regular function GL_n(k)\to k.

(iii) For x in GL_n(k), f in R we define L_xf:GL_n(k)\to k by (L_xf)(g)=f(x^{-1}g).
Let V_f be the subspace of R spanned by L_xf for various x in G. 
Deduce from (i) that V_f is finite dimensional.
Show that x:f'\to L_xf' is a group homomorphism GL_n(k)\to GL(V_f).

ADDED APR.12.: It is OK to prove instead of (i) the following easier statement:

(i') For any f\in R there exist  f_1,f'_1,f_2,f'_2,...,f_s,f'_s in R such that
 f(xy)=f_1(x)f'_1(y)+f_2(x)f'_2(y)+...+f_s(x)f'_s(y)
 for any x,y in GL_n(k).